Sensitivity analysis for attributable effects in case2 studies

The case$^2$ study, also referred to as the case-case study design, is a valuable approach for conducting inference for treatment effects. Unlike traditional case-control studies, the case$^2$ design compares treatment in cases of concern (the first type of case) to other cases (the second type of case). One of the quantities of interest is the attributable effect for the first type of case-that is, the number of the first type of case that would not have occurred had the treatment been withheld from all units. In some case$^2$ studies, a key quantity of interest is the attributable effect for the first type of case. Two key assumptions that are usually made for making inferences about this attributable effect in case$^2$ studies are (1) treatment does not cause the second type of case, and (2) the treatment does not alter an individual's case type. However, these assumptions are not realistic in many real-data applications. In this article, we present a sensitivity analysis framework to scrutinize the impact of deviations from these assumptions on inferences for the attributable effect. We also include sensitivity analyses related to the assumption of unmeasured confounding, recognizing the potential bias introduced by unobserved covariates. The proposed methodology is exemplified through an investigation into whether having violent behavior in the last year of life increases suicide risk using the 1993 National Mortality Followback Survey dataset.